January 3-11, 2015

We will emphasize and study the role of equivariant harmonic maps as a crucial link between Higgs bundles, geometric structures and representation varieties. After preliminary talks on harmonic maps, Higgs bundles and the geometry of symmetric spaces, further topics will include: Corlette's theorem on the existence of equivariant harmonic maps, relationships between harmonic maps and integrable systems, and harmonic maps to singular spaces. Subsequent talks will be based on current developments in the theory of harmonic maps, Higgs bundles and geometric structures. In particular, applications to studying minimal surfaces in symmetric spaces, deformation spaces of OPERS, the geometry of Anosov representations with quasi-Fuchsian representations and Hitchin representations serving as illustrative examples, and asymptotic analysis of such deformation spaces. The overarching theme may be summarized as follows: how does one unpack the data of Higgs bundles, and via a careful study of harmonic maps, start to understand the geometry of surface group representations in a more explicit way.

The workshop will consist of whiteboard talks by the participants on the following topics. Speakers will be alotted 2.5 hours per talk. Speakers will be asked to submit a 5-6 page summary, clicking on the each title below will lead you to that talks summary.

#### 1. Rob Maschal: Higgs bundles background

An outline of the nonabelian Hodge correspondence with focus on Higgs bundles, especially the Hitchin component and $SL(2,\mathbb{C})$.

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#### 2. Tengren Zhang: Geometry of Symmetric and Homogeneous spaces

Semisimple Lie groups: Cartan decompositions, Riemannian geometry, and boundaries of the associated symmetric spaces.

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#### 3. Jérémy Toulisse: Existence theory for harmonic metrics.

Corlette's theorem

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#### 4. Semin Kim: Harmonic maps to $\mathbb{R}$-trees and Morgan Shalen compactification

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#### 5. Andrew Sanders: Background on Labourie's conjecture on existence and uniqueness of minimal surfaces for Hitchin representations.

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#### 6. Marco Spinaci: Labourie's recent paper: Cyclic surfaces and Hitchin components in rank 2.

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#### 7. Qiongling Li: Background on harmonic maps to metric spaces and survey of Katzarkov, Noll, Pandit, and Simpson's paper: Harmonic maps to Buildings and Singular perturbation theory.

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#### 8. Jorge Acosta: OPERS and complex projective structures.

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#### 9. Brian Collier: The relationship between Integrable systems and Harmonic maps with emphasis on the Toda lattice.

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#### 10. Andy Huang: The relationship between Integrable systems and Harmonic maps with emphasis on the Toda lattice.

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#### 11. Brice Loustau: Minimal surfaces in $\mathbb{H}^3$ and quasi-Fuchsian representations.

Minimal surfaces as equivariant harmonic maps, Taubes moduli space of minimal hyperbolic germs, explicit examples of corresponding $SL(2,\mathbb{C})$ Higgs bundles.

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#### 12. Jakob Blaavand: Exposition of Ends of the moduli space of Higgs bundles by Mazzeo, Swoboda, Weiss, Witt.

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#### 13. Laura Fredrickson: Survey of Taubes's paper $PSL(2,\mathbb{C})$-connections with $L^2$ bounds on curvature.

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#### 14. Daniele Alessandrini: Branched hyperbolic surfaces and nonmaximal $SL(2,\mathbb{R})$ representations/Higgs bundles.

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9am | 2:30pm | 8pm | |

Saturday | Robert Maschal | ||

Sunday | Tengren Zhang | Jeremy Toulisse | |

Monday | Semin Kim | Andy Sanders | |

Tuesday | Marco Spinaci | Qiongling Li | |

Wednesday | Jorge Acosta | ||

Thursday | Laura Fredrickson | Brice Loustau | |

Friday | Jakob Blaavand | Andy Huang | |

Saturday | Brian Collier | Daniele Alessandrini |

#### Location:

The workshop will take place January 3-11, 2015 in a large cabin in the mountains of North Carolina near Asheville.

#### Transportation:

Transportation between Asheville airport and the cabin will be provided on Saturday, January 3, and Sunday, January 11. Please email your itinerary to the organizers as soon as it becomes available.

#### Funding:

Full funding for the workshop is provided by the GEAR network. Invited participants will be reimbursed for travel (see your email). Please email your receipts to the organizers prior to the workshop or bring printed copies with you.